- Research Article
- Open Access
Global Behaviors and Optimal Harvesting of a Class of Impulsive Periodic Logistic Single-Species System with Continuous Periodic Control Strategy
© JinRongWang et al. 2008
Received: 30 September 2008
Accepted: 17 December 2008
Published: 8 February 2009
Global behaviors and optimal harvesting of a class of impulsive periodic logistic single-species system with continuous periodic control strategy is investigated. Four new sufficient conditions that guarantee the exponential stability of the impulsive evolution operator introduced by us are given. By virtue of exponential stability of the impulsive evolution operator, we present the existence, uniqueness and global asymptotical stability of periodic solutions. Further, the existence result of periodic optimal controls for a Bolza problem is given. At last, an academic example is given for demonstration.
In population dynamics, the optimal management of renewable resources has been one of the interesting research topics. The optimal exploitation of renewable resources, which has direct effect on their sustainable development, has been paid much attention [1–3]. However, it is always hoped that we can achieve sustainability at a high level of productivity and good economic profit, and this requires scientific and effective management of the resources.
Single-species resource management model, which is described by the impulsive periodic logistic equations on finite-dimensional spaces, has been investigated extensively, no matter how the harvesting occurs, continuously [1, 4] or impulsively [5–7]. However, the associated single-species resource management model on infinite-dimensional spaces has not been investigated extensively.
Since the end of last century, many authors including Professors Nieto and Hernández pay great attention on impulsive differential systems. We refer the readers to [8–22]. Particulary, Doctor Ahmed investigated optimal control problems [23, 24] for impulsive systems on infinite-dimensional spaces. We also gave a series of results [25–34] for the first-order (second-order) semilinear impulsive systems, integral-differential impulsive system, strongly nonlinear impulsive systems and their optimal control problems. Recently, we have investigated linear impulsive periodic system on infinite-dimensional spaces. Some results [35–37] including the existence of periodic -mild solutions and alternative theorem, criteria of Massera type, asymptotical stability and robustness against perturbation for a linear impulsive periodic system are established.
On infinite-dimensional spaces, where denotes the population number of isolated species at time and location , is a bounded domain and , operator . The coefficients , are sufficiently smooth functions of in , where , and , . Denoting , , then = . is related to the periodic change of the resources maintaining the evolution of the population and the periodic control policy , where is a suitable admissible control set. Time sequence and as , denote mutation of the isolate species at time where .
where is Borel measurable, is continuous, and nonnegative and denotes the -periodic -mild solution of system (1.1) at location and corresponding to the control . The Bolza problem ( ) is to find such that for all
Suppose that , , and is the least positive constant such that there are s in the interval and where , . The first equation of system (1.1) describes the variation of the population number of the species in periodically continuous controlled changing environment. The second equation of system (1.1) shows that the species are isolated. The third equation of system (1.1) reflects the possibility of impulsive effects on the population.
where denotes the -periodic -mild solution of system (1.3) corresponding to the control . The Bolza problem (P) is to find such that for all The investigation of the system (1.3) cannot only be used to discuss the system (1.1), but also provide a foundation for research of the optimal control problems for semilinear impulsive periodic systems. The aim of this paper is to give some new sufficient conditions which will guarantee the existence, uniqueness, and global asymptotical stability of periodic -mild solutions for system (1.3) and study the optimal control problems arising in the system (1.3).
The paper is organized as follows. In Section 2, the properties of the impulsive evolution operator are collected. Four new sufficient conditions that guarantee the exponential stability of the are given. In Section 3, the existence, uniqueness, and global asymptotical stability of -periodic -mild solution for system (1.3) is obtained. In Section 4, the existence result of periodic optimal controls for the Bolza problem (P) is presented. At last, an academic example is given to demonstrate our result.
2. Impulsive Periodic Evolution Operator and It's Stability
Let be a Banach space, denotes the space of linear operators on ; denotes the space of bounded linear operators on . is the Banach space with the usual supremum norm. Denote and define is continuous at , is continuous from left and has right-hand limits at and .
We introduce assumption [H1].
- (1)By assumption [H1.1], there exists a constant such that . Using assumption [H1.3], it is obvious that , for . (2) By the definition of -semigroup and the construction of , one can verify the result immediately. (3) By assumptions [H1.2], [H1.3], and elementary computation, it is easy to obtain the result. (4) For , , by virtue of (3) again and again, we arrive at
This completes the proof.
In order to study the asymptotical properties of periodic solutions, it is necessary to discuss the exponential stability of the impulsive evolution operator . We first give the definition of exponential stable for .
Lemma 2.3 (see [38, Lemma 7.2.1]).
and the boundedness of , are convergent, that for every and fixed . This shows that is bounded for each and fixed and hence, by virtue of uniform boundedness principle, there exists a constant such that for all . Let denote the operator given by , and is fixed. Clearly, is defined every where on and by assumption it maps and it is a closed operator. Hence, by closed graph theorem, it is a bounded linear operator from to . Thus, there exits a constant such that for all and , is fixed.
3. Periodic Solutions and Global Asymptotical Stability
In addition to assumption [H1], we make the following assumptions:
Theorem 3.2 .A.
4. Existence of Periodic Optimal Harvesting Policy
In this section, we discuss existence of periodic optimal harvesting policy, that is, periodic optimal controls for optimal control problems arising in systems governed by linear impulsive periodic system on Banach space.
is strongly continuous.
Now we can give the following results on existence of periodic optimal controls for Bolza problem (P).
Last, an academic example is given to illustrate our theory.
where denotes time, denotes age, is called age density function, and are positive constants, is a bounded measurable function, that is, . denotes the age-specific death rate, denotes the age density of migrants, and denotes the control. The admissible control set .
By the fact that the operator is an infinitesimal generator of a -semigroup (see [39, Example 2.21]) and [38, Theorem 4.2.1], then is an infinitesimal generator of a -semigroup since the operator is bounded.
By Lemma 2.4, for , is exponentially stable. Now, all the assumptions are met in Theorems 3.2 and 4.3, our results can be used to system (5.1). Thus, system (5.1) has a unique -periodic -mild solution which is globally asymptotically stable and there exists a periodic control such that for all
The results show that the optimal population level is truly the periodic solution of the considered system, and hence, it is globally asymptotically stable. Meanwhile, it implies that we can achieve sustainability at a high level of productivity and good economic profit by virtue of scientific, effective, and continuous management of the resources.
This work is supported by Natural Science Foundation of Guizhou Province Education Department (no. 2007008). This work is also supported by the undergraduate carve out project of Department of Guiyang Science and Technology (2008, no. 15-2).
- Clark CW: Mathematical Bioeconomics: The Optimal Management of Renewable Resources, Pure and Applied Mathematics. John Wiley & Sons, New York, NY, USA; 1976:xi+352.Google Scholar
- Song X, Chen L: Optimal harvesting and stability for a two-species competitive system with stage structure. Mathematical Biosciences 2001,170(2):173-186. 10.1016/S0025-5564(00)00068-7MathSciNetView ArticleMATHGoogle Scholar
- Marzollo R (Ed): Periodic Optimization. Springer, New York, NY, USA; 1972.MATHGoogle Scholar
- Fan M, Wang K: Optimal harvesting policy for single population with periodic coefficients. Mathematical Biosciences 1998,152(2):165-177. 10.1016/S0025-5564(98)10024-XMathSciNetView ArticleMATHGoogle Scholar
- Bainov DD, Simeonov PS: Impulsive Differential Equations: Periodic Solutions and Applications, Pitman Monographs and Surveys in Pure and Applied Mathematics. Volume 66. Longman Scientific & Technical, Harlow, UK; 1993:x+228.MATHGoogle Scholar
- Xiao YN, Cheng DZ, Qin HS: Optimal impulsive control in periodic ecosystem. Systems & Control Letters 2006,55(7):558-565. 10.1016/j.sysconle.2005.12.003MathSciNetView ArticleMATHGoogle Scholar
- Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. Volume 6. World Scientific, Teaneck, NJ, USA; 1989:xii+273.View ArticleGoogle Scholar
- Nieto JJ: An abstract monotone iterative technique. Nonlinear Analysis: Theory, Methods & Applications 1997,28(12):1923-1933. 10.1016/S0362-546X(97)89710-6MathSciNetView ArticleMATHGoogle Scholar
- Yan J, Zhao A, Nieto JJ: Existence and global attractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systems. Mathematical and Computer Modelling 2004,40(5-6):509-518. 10.1016/j.mcm.2003.12.011MathSciNetView ArticleMATHGoogle Scholar
- Jiao J-J, Chen L-S, Nieto JJ, Angela T: Permanence and global attractivity of stage-structured predator-prey model with continuous harvesting on predator and impulsive stocking on prey. Applied Mathematics and Mechanics 2008,29(5):653-663. 10.1007/s10483-008-0509-xMathSciNetView ArticleMATHGoogle Scholar
- Zeng G, Wang F, Nieto JJ: Complexity of a delayed predator-prey model with impulsive harvest and Holling type II functional response. Advances in Complex Systems 2008,11(1):77-97. 10.1142/S0219525908001519MathSciNetView ArticleMATHGoogle Scholar
- Wang W, Shen J, Nieto JJ: Permanence and periodic solution of predator-prey system with Holling type functional response and impulses. Discrete Dynamics in Nature and Society 2007, 2007:-15.Google Scholar
- Zhang H, Chen L, Nieto JJ: A delayed epidemic model with stage-structure and pulses for pest management strategy. Nonlinear Analysis: Real World Applications 2008,9(4):1714-1726. 10.1016/j.nonrwa.2007.05.004MathSciNetView ArticleMATHGoogle Scholar
- Ahmad B, Nieto JJ: Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 2008,69(10):3291-3298. 10.1016/j.na.2007.09.018MathSciNetView ArticleMATHGoogle Scholar
- Jiang D, Yang Y, Chu J, O'Regan D: The monotone method for Neumann functional differential equations with upper and lower solutions in the reverse order. Nonlinear Analysis: Theory, Methods & Applications 2007,67(10):2815-2828. 10.1016/j.na.2006.09.042MathSciNetView ArticleMATHGoogle Scholar
- Li Y: Global exponential stability of BAM neural networks with delays and impulses. Chaos, Solitons & Fractals 2005,24(1):279-285.MathSciNetView ArticleMATHGoogle Scholar
- De la Sen M: Stability of impulsive time-varying systems and compactness of the operators mapping the input space into the state and output spaces. Journal of Mathematical Analysis and Applications 2006,321(2):621-650. 10.1016/j.jmaa.2005.08.038MathSciNetView ArticleMATHGoogle Scholar
- Yu L, Zhang J, Liao Y, Ding J: Parameter estimation error bounds for Hammerstein nonlinear finite impulsive response models. Applied Mathematics and Computation 2008,202(2):472-480. 10.1016/j.amc.2008.01.002MathSciNetView ArticleMATHGoogle Scholar
- Jankowski T: Positive solutions to second order four-point boundary value problems for impulsive differential equations. Applied Mathematics and Computation 2008,202(2):550-561. 10.1016/j.amc.2008.02.040MathSciNetView ArticleMATHGoogle Scholar
- Hernández EM, Pierri M, Goncalves G: Existence results for an impulsive abstract partial differential equation with state-dependent delay. Computers & Mathematics with Applications 2006,52(3-4):411-420. 10.1016/j.camwa.2006.03.022MathSciNetView ArticleMATHGoogle Scholar
- Hernández EM, Rabello M, Henríquez HR: Existence of solutions for impulsive partial neutral functional differential equations. Journal of Mathematical Analysis and Applications 2007,331(2):1135-1158. 10.1016/j.jmaa.2006.09.043MathSciNetView ArticleMATHGoogle Scholar
- Hernández EM, Sakthivel R, Aki ST: Existence results for impulsive evolution differential equations with state-dependent delay. Electronic Journal of Differential Equations 2008,2008(28):1-11.MathSciNetMATHGoogle Scholar
- Ahmed NU: Some remarks on the dynamics of impulsive systems in Banach spaces. Dynamics of Continuous, Discrete and Impulsive Systems. Series A 2001,8(2):261-274.MathSciNetMATHGoogle Scholar
- Ahmed NU, Teo KL, Hou SH: Nonlinear impulsive systems on infinite dimensional spaces. Nonlinear Analysis: Theory, Methods & Applications 2003,54(5):907-925. 10.1016/S0362-546X(03)00117-2MathSciNetView ArticleMATHGoogle Scholar
- Xiang X, Ahmed NU: Existence of periodic solutions of semilinear evolution equations with time lags. Nonlinear Analysis: Theory, Methods & Applications 1992,18(11):1063-1070. 10.1016/0362-546X(92)90195-KMathSciNetView ArticleMATHGoogle Scholar
- Xiang X: Optimal control for a class of strongly nonlinear evolution equations with constraints. Nonlinear Analysis: Theory, Methods & Applications 2001,47(1):57-66. 10.1016/S0362-546X(01)00156-0MathSciNetView ArticleMATHGoogle Scholar
- Sattayatham P, Tangmanee S, Wei W: On periodic solutions of nonlinear evolution equations in Banach spaces. Journal of Mathematical Analysis and Applications 2002,276(1):98-108. 10.1016/S0022-247X(02)00378-5MathSciNetView ArticleMATHGoogle Scholar
- Xiang X, Wei W, Jiang Y: Strongly nonlinear impulsive system and necessary conditions of optimality. Dynamics of Continuous, Discrete & Impulsive Systems. Series A 2005,12(6):811-824.MathSciNetMATHGoogle Scholar
- Wei W, Xiang X: Necessary conditions of optimal control for a class of strongly nonlinear impulsive equations in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2005,63(5–7):53-63.View ArticleGoogle Scholar
- Wei W, Xiang X, Peng Y: Nonlinear impulsive integro-differential equations of mixed type and optimal controls. Optimization 2006,55(1-2):141-156. 10.1080/02331930500530401MathSciNetView ArticleMATHGoogle Scholar
- Xiang X, Wei W: Mild solution for a class of nonlinear impulsive evolution inclusions on Banach space. Southeast Asian Bulletin of Mathematics 2006,30(2):367-376.MathSciNetMATHGoogle Scholar
- Yu X, Xiang X, Wei W: Solution bundle for a class of impulsive differential inclusions on Banach spaces. Journal of Mathematical Analysis and Applications 2007,327(1):220-232. 10.1016/j.jmaa.2006.03.075MathSciNetView ArticleMATHGoogle Scholar
- Peng Y, Xiang X, Wei W: Nonlinear impulsive integro-differential equations of mixed type with time-varying generating operators and optimal controls. Dynamic Systems and Applications 2007,16(3):481-496.MathSciNetMATHGoogle Scholar
- Peng Y, Xiang X: Second order nonlinear impulsive time-variant systems with unbounded perturbation and optimal controls. Journal of Industrial and Management Optimization 2008,4(1):17-32.MathSciNetView ArticleMATHGoogle Scholar
- Wang J: Linear impulsive periodic system on Banach spaces. Proceedings of the 4th International Conference on Impulsive and Hybrid Dynamical Systems, July 2007, Nanning, China 5: 20-25.Google Scholar
- Wang J, Xiang X, Wei W: Linear impulsive periodic system with time-varying generating operators on Banach space. Advances in Difference Equations 2007, 2007:-16.Google Scholar
- Wang J, Xiang X, Wei W, Chen Q:Existence and global asymptotical stability of periodic solution for the -periodic logistic system with time-varying generating operators and -periodic impulsive perturbations on Banach spaces Discrete Dynamics in Nature and Society 2008, 2008:-16.Google Scholar
- Ahmed NU: Semigroup Theory with Applications to Systems and Control, Pitman Research Notes in Mathematics Series. Volume 246. Longman Scientific & Technical, Harlow, UK; 1991:x+282.Google Scholar
- Luo Z-H, Guo B-Z, Morgül O: Stability and Stabilization of Infinite Dimensional Systems with Applications, Communications and Control Engineering Series. Springer, London, UK; 1999:xiv+403.View ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.